Doubly heavy tetraquark states $cc\bar{u}\bar{d}$ and $bb\bar{u}\bar{d}$
Inspired by the recent observation of a very narrow state, called $T_{cc}^+$, by the LHCb collaboration, the possible bound states and low-lying resonance states of the doubly heavy tetraquark states $cc\bar{u}\bar{d}$ ($T_{cc}$) and $bb\bar{u}\bar{d}$ ($T_{bb}$) are searched in the framework of chiral quark model with an accurate few body method, Gaussian expansion method. The real scaling method is also applied to identify the genuine resonance states. In the calculation, the meson-meson structure, diquark-antidiquark structure, as well as their coupling are all considered. The numerical results show that (i) for $T_{cc}$ and $T_{bb}$, only $I(J^P)=0(1^+)$ states are bound in different quark structures. The binding energy varies from a few MeV for meson-meson structure to over 100 MeV for diquark-antidiquark structure. For example for $T_{cc}$, in meson-meson structure, there exists a weakly bound molecule $DD^*$ state around 3841.4 MeV, 1.8 MeV below the $D^0D^{*+}$, which may be a good candidate of the observed state by LHCb; however in diquark-antidiquark structure, a deeper bound sate with mass 3700.9 MeV is obtained; when considering the structure mixing, the energy of system decreases to 3660.7 MeV and the shallow bound state disappears. (ii) Besides bound states, several resonance states for $T_{QQ}(Q=c, b)$ with $I(J^P)=1(0^+), 1(1^+), 1(2^+), 0(1^+)$ are proposed.